| L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 2·13-s − 14-s + 16-s + 6·17-s − 4·19-s + 20-s + 25-s − 2·26-s + 28-s + 6·29-s − 4·31-s − 32-s − 6·34-s + 35-s + 2·37-s + 4·38-s − 40-s − 6·41-s + 8·43-s + 12·47-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.392·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265534707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265534707\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47203449948790661999657678873, −9.787769285156101965157969205963, −8.800046649726438641176234759863, −8.134234480169490737550982107259, −7.16665739633458789232295323445, −6.15797813375789816669713446161, −5.29787033315398374470104513854, −3.88231405811014601850145796336, −2.51414806834535899531438368517, −1.18226526293597327102151626443,
1.18226526293597327102151626443, 2.51414806834535899531438368517, 3.88231405811014601850145796336, 5.29787033315398374470104513854, 6.15797813375789816669713446161, 7.16665739633458789232295323445, 8.134234480169490737550982107259, 8.800046649726438641176234759863, 9.787769285156101965157969205963, 10.47203449948790661999657678873
